# TRIANGLE CONSTRUCTION

The design for the interior of the dome is made up of six hexagonal layers, each layer surmounting the one before. The creation of each layer is based on a 360o circle with the hexagon inscribed within the circle. The six points for the base layer of the hexagon are placed counter-clockwise along the circle at the following angles 0o, 60o, 120o, 180o, 240o, and 300o relative to the positive x-axis. To compute the point (x,y) along a circle at a given angle t, use the formula (x,y) = (rcost, rsint).

The z-coordinate is the height of the layer above the origin as measured on the side-elevation. The base level has a z-coordinate of 0 since the height at that level is 0.00 cm. The radius is computed for each layer using its height as v in the equation for the two cones. So for example, the radius for level 1 is 5.38 cm since its height is 1.70 cm. This is used to compute the first hexagon layer. In order to compute one triangle layer one needs to know the points along the circle for two hexagons: the one for the layer sought and the one above. For example, when computing the base level, one needs the angles mentioned above and the angles for the first triangle layer which are 30o, 90o, 150o, 210o, 270o, and 330o. The triangles in this layer are formed by connecting the point at 0o on the base layer with the points at 30o and 330o on the first layer, connecting the point at 60o on the base layer with the points at 30o and 90o on the first layer, and so on. Both these sets of angles are standard in computing the other five layers. This is because hexagons in alternate layers have vertices at the same angles. Thus, the second triangle layer uses the same angles as the base level and the third layer uses the same angles as the first. This pattern continues for all six layers.

 Optical Illusion & Projection in Domes: A Study of Guarino Guarini's Santissima Sindone